Memory tips for liberal arts mathematics knowledge points in the second grade of high school

#高二# Introduction The second grade of high school has two major characteristics: 1. The teaching progress is fast. It takes one year to complete the two-year course. 2. The freshness of the first year of high school is over, and the college entrance examination is still far away. It is easiest to play crazy and go far away. It leads to: a period of psychological confusion, a period of slow academic progress, a period of loose self-restraint, a period of being easily led astray, and a period of screening when the waves are rushing through the sand. Therefore, it is of great significance and urgency to face the challenges of the sophomore year, recognize the sophomore self, and the tasks of the sophomore. The high school sophomore channel has compiled "Memory Tips for Liberal Arts Mathematics Knowledge Points for Sophomore High School Students" for you. We hope it will be helpful to your study!

1. "Sets and Functions"

Contents include intersection and complement sets, as well as power pair functions. The properties of odd-even and increase/decrease are most obvious when observing images.

Composite functional expression appears, and the property multiplication rule is identified. If you want to prove it in detail, you must grasp the definition.

Exponential and logarithmic functions are inverse functions of each other. A positive number with a base other than 1 means an increase or decrease on both sides of 1.

The function domain is easy to find. The denominator cannot be equal to 0, the even square root must be non-negative, and there is no logarithm between zero and negative numbers;

The angles of the tangent function are not straight, and the angles of the cotangent function are not flat; for other real function sets, the intersection can be found in various situations.

The two mutually inverse functions have the same monotonic properties; the images are axially symmetrical to each other, and Y=X is the axis of symmetry;

The solution is very regular, and the inverse solution is the definition of the substitution element Domain; the domain of the inverse function, the value domain of the original function.

The properties of power functions are easy to remember, and exponentiation reduces fractions; the properties of functions depend on the exponent, odd functions with odd mothers and odd children,

Even functions with even mothers and even children, and non-odd and even functions with even mothers ;In the first quadrant of the image, the increase or decrease of the function is positive or negative.

2. "Trigonometric Functions"

Trigonometric functions are functions, quadrant symbol coordinates note. The function graph is the unit circle, and the odd and even periods increase or decrease.

The same angle relationship is very important and is required for simplification proof. At the vertex of the regular hexagon, cut from top to bottom chord;

The number 1 is written in the center to connect the vertex triangles; the sum of the squares of the downward triangles, the reciprocal relationship is the diagonal,

The vertex is arbitrary One function is equal to the eradication of the next two. The induction formula is good. After turning the negative into the positive, it becomes large and small.

It becomes the tax angle and is easy to look up in the table. The simplification proof is indispensable. Half of two is an integer multiple, and the remainder remains unchanged when converted to odd numbers.

Treat the latter as an acute angle, and judge the original function of the sign. The cosine value of the sum of two angles can be easily evaluated by converting it into a single angle.

The product of cosine minus the product of sine can be transformed into multiple formulas by changing angles. The sum and difference products must have the same name, and the complementary angles must have the same name.

Calculate the proof angle first, pay attention to the name of the structural function, keep the basic quantities unchanged, and change the complexity to simplicity.

Guided by the inverse principle, ascending powers, descending times and difference products. Proof of conditional equality, equation thinking guides the way.

The universal formula is unusual, and it is the first to be transformed into a rational formula. The formula can be used smoothly and conversely, and the deformation and clever use can be used;

1 plus cosine is like cosine, 1 minus cosine is like sine, when the power is raised to the first degree, the angle is halved, and when the power is raised to the lower degree, it is the norm;

The essence of the inverse function of trigonometric functions is to find the angle. First find the value of the trigonometric function, and then determine the value range of the angle;

Using the right triangle, the image is intuitive and easy to change the name. The equations of simple triangles can be transformed into is the simplest solution set;

3. "Inequalities"

The way to solve inequalities is to use the properties of functions. The opposite refers to irrational inequalities, which are transformed into rational inequalities.

From higher to lower generations, the step-by-step transformation must be equivalent. The mutual conversion between numbers and shapes is very helpful in solving problems.

The method of proving inequalities is powerful in the properties of real numbers. Compare the difference with 0, and compare the difference with 1.

Analyze direct difficulties well, and have clear and comprehensive ideas. For non-negativity, basic expressions are often used. If it is difficult to be positive, then proof by contradiction is used.

There are also important inequalities and mathematical induction. Graphical functions to help, drawing modeling construction method.

IV. "Sequence"

Arithmetic two-number sequence, general formula N terms sum.

Two finites are used to find the limit, and the order of the four operations is changed.

Sequence problems are subject to many changes, and the equations must be reduced to the overall calculation. It is more difficult to sum up a sequence. It can be calculated by using the dislocation destructive clever transformation,

using the Gaussian method to make up for the shortcomings, and the formula for summation of split terms. Inductive thinking is very good. It is easy to write a program to think:

One calculation, two observations and three associations, guessing and proving are indispensable. There is also mathematical induction, and the proof steps are programmed:

First verify and then assume, add 1 from K to K, the inference process must be detailed, and the induction principle must be used to confirm.

5. "Complex Numbers"

Once the imaginary unit i comes out, the number set is expanded to complex numbers. A complex number is a pair of numbers, with the real and imaginary parts of the horizontal and vertical coordinates.

Corresponds to the point on the complex plane, and the origin is connected with it to form an arrow. The arrow shaft is in the positive direction of the X-axis, and the resulting angle is the spoke angle.

The length of the arrow shaft is the mold, and numbers and shapes are often combined. Try converting algebraic geometric trigonometric formulas into each other.

The essence of algebraic operations includes polynomial operations. The positive integer degree of i has four numerical periods.

Some important conclusions can be obtained by memorizing them and using them skillfully. The ability to transform reality into reality is great, and complex numbers can be transformed if they are equal.

Use equations to think about solutions and pay attention to the overall substitution technique. Looking at the geometric operation diagram, addition of parallelograms, subtraction of trigonometric rules; operations of multiplication and division include rotation in the reverse and forward direction, and expansion and contraction of the length of the whole year.

The calculation of trigonometric forms requires the identification of arguments and modules. Using De Moivre's formula, it is very convenient to perform exponentiation and square root.

The argument operation is very strange, the sum and difference are obtained by the product quotient. Four properties are inseparable, equality and modulus and ***yoke,

The two will not be real numbers, so comparison is indispensable. Complex real numbers are very closely related, and we must pay attention to their essential differences.

6. "Permutation, Combination, and Binomial Theorem"

The two principles of addition and multiplication are the rules that run through the whole process. It is combination that has nothing to do with order, and it is permutation that requires order.

Two formulas are of essence, two ideas and methods. To summarize the permutations and combinations, application problems must be transformed.

When arranged and combined, it is common sense to choose first and second. Special elements and locations should be considered first.

Don’t emphasize too much, don’t omit too much, think too much, and inserting blanks is a skill. Permutation and combination of identities, definition and proof modeling test.

Regarding the binomial theorem, Yang Hui’s triangle in China. Two properties, two formulas, function assignment transformation formula.

7. "Solid Geometry"

The trinity of points, lines and surfaces is represented by the cylinder, cone and billiards. Distances start from points, and angles start from lines.

Vertical parallelism is the key point, and the proof requires clarifying the concept. Lines, lines, planes, planes, and three pairs appear in cycles.

Equations are solved in a holistic way, and then reduced to conscious and dynamic cuts and complements. Before calculation, it is necessary to prove and draw the removed graph.

Three-dimensional geometric auxiliary lines, commonly used vertical lines and planes. The concept of projection is very important and is the most critical to solving problems.

Dihedral angles of different planes and volume projection formulas are live. The axiom property of three vertical lines can solve a large number of problems.

8. "Plane Analytical Geometry"

Directed line segments, straight circles, elliptical hyperbolic parabolas, parametric equations, polar coordinates, the combination of numbers and shapes is called a model.

Descartes’ pairs of viewpoints, pairs of points and ordered real numbers, correspond to each other and create a new approach to geometry.

The two ideas complement each other, and the reduction idea takes the lead; it is said that the method of undetermined coefficients is actually the idea of ??a system of equations.

The three types are integrated, draw a curve to find an equation, draw a curve given the equation, and judge the position relationship of the curve.

The four tools are magic weapons, coordinate thinking parameters are good; plane geometry cannot be lost, and rotation and transformation can be calculated with complex numbers.

Analytical geometry is geometry, and you can’t live without getting carried away. Graphics are intuitive and mathematical, and mathematics is essentially morphology.